1. (2 pt) Betelgeuse (the red star in Orion's armpit) has a resolvable disk with a diameter of 0.054 arc seconds which is just barely discernable by Hubble Space Telescope. If it is at a distance of 425 light years, calculate its actual diameter in km. (A helpful website is http://www.astronomycafe.net/qadir/q2278.html) but DON'T just take his number of 734 solar diameters and change that to km... I want you to show you steps in the small-angle calculation: d/D = angle in radians if d/D is small.
2. The "apparent magnitude" of a star is a logarithmic measure of how bright a star appears. The brightest stars have magnitudes of -1 to 0, and the dimmest naked-eye stars have magnitudes of 5 or 6. Each 5 steps in magnitude corresponds to a factor of 100 in brightness. Thus if one star is a magnitude of 12 and the other is a magnitude of 4, then difference of magnitudes is 8, which means that the ratio of apparent brightness is greater than 100 (which would be 5 steps) and closer to 10000 (which would be 10 steps). Do to it exactly, brightness of star 2 divided by the brightness of star 1 is given by (10**((m2 - m1)/2.5)). What is the exact brightness ratio for these two stars?
3. By using a large aperture, we collect more light from a star, allowing us to see much dimmer stars. If the Rice telescope is 16 inch diameter and your eye pupil is only .25 inch, what is the ratio of area of light collected? This then can allow us to see dimmer stars. If with your naked eye you can see a 5th magnitude star, how dim a star can we see from the Rice observatory - that is, what is the magnitude of the dimmest star you can see from Rice? (the ratio in brightness is the ratio of AREA not diameter!)
4. We also can see dimmer stars by integrating the light over time, by having long-time exposures for CCD's (previously, for camera film). If you can have a 4-hour time exposure on an earth-based telescope, what is the ratio of time between that and your eye integrating time (1/30 sec)? How many magnitudes in brightness does that correspond to?
5. The "Absolute Magnitude" is the apparent magnitude of a star if you move it to a distance of 10 pc (1 pc = 3.262 ly = 3.09 E16 m). The "Distance Modulus", (m - M), is the apparent magnitude m minus the absolute magnitude M, and is a logarithmic measure of distance. m - M = 5 log (r/1pc) - 5 where r is measured in pc. Check: for r = 10 pc, log r = 1, so m - M = 0. If the distance to Betelgeuse is 425 pc, what is its distance modulus?
6. If Betelgeuse has an apparent magnitude of 0.5, what would its absolute magnitude be? Does this agree with any textbooks you find?
7. Estimates of the value of the Hubble constant H, are converging to about 71 km/s per Mpc (megaparsec) (for an interesting history see http://cfa-www.harvard.edu/~huchra/hubble/. This means, for a galaxy which is 1 Mpc away, its speed away from us is 71 km/s, and farther away galaxies are receding faster. For this value of H, what is the age of the universe if the expansion has been uniform in time? (1 / H, but watch your units!). At that age, what is the knowable size of the universe (the Hubble time times the speed of light). (Give in MPc as well as km). Is that the same as the distance at which the recession speed equals the speed of light? Please note: you *can* get a doppler shift z = delta (wavelength) / wavelength for very distant galaxies greater than one! In that case, the simple doppler formula we did before has to be modified for high speeds. In that case, z = delta L / Lo = -1 + SQRT ((1 - v/c)/(1 + v/c)). This formula reduces to z = v/c for small v's.
Last updated 2/21/2006